Therefore even when the sparse graphs as in Fig. REF (e)-(g) require a few colors the p-bits with 32 or 498 neighbors introduce large synapse (addition) delays creating a severe bottleneck for how fast the clocks can be operated. To overcome this limitation we develop exact sparsification techniques to remove the nodes with a large number of neighbors without changing the structure of the optimization problem. The main idea in this approach is to split a given p-bit representation between two p-bits coupled by a ferromagnetic (\(J>\) 0) interaction which we call a COPY gate [1]}, [2]}, [3]}. The ferromagnetic interaction ensures that at the end of an annealing schedule (high \(\beta \) ), the ground states of the split and fused models are identical. We show a formal proof in the Supplementary Information Section REF .
